cp's OEIS Frontend

A113350 Triangle Q, read by rows, such that Q^2 transforms column k of Q^2 into column k+1 of Q^2, so that column k of Q^2 equals column 0 of Q^(2*k+2), where Q^2 denotes the matrix square of Q.

%I A113350 #5 Jun 13 2017 22:51:36
%S A113350 1,2,1,5,4,1,19,22,6,1,113,166,51,8,1,966,1671,561,92,10,1,10958,
%T A113350 21510,7726,1324,145,12,1,156700,341463,129406,23010,2575,210,14,1,
%U A113350 2727794,6496923,2572892,471724,53935,4434,287,16,1
%N A113350 Triangle Q, read by rows, such that Q^2 transforms column k of Q^2 into column k+1 of Q^2, so that column k of Q^2 equals column 0 of Q^(2*k+2), where Q^2 denotes the matrix square of Q.
%F A113350 Let [Q^m]_k denote column k of matrix power Q^m,
%F A113350 so that triangular matrix Q may be defined by
%F A113350 [Q]_k = [P^(2*k+2)]_0, for k>=0, where
%F A113350 the dual triangular matrix P = A113340 is defined by
%F A113350 [P]_k = [P^(2*k+1)]_0, for k>=0.
%F A113350 Then, amazingly, powers of P and Q satisfy:
%F A113350 [P^(2*j+1)]_k = [P^(2*k+1)]_j,
%F A113350 [P^(2*j+2)]_k = [Q^(2*k+1)]_j,
%F A113350 [Q^(2*j+2)]_k = [Q^(2*k+2)]_j,
%F A113350 for all j>=0, k>=0.
%F A113350 Also, we have the column transformations:
%F A113350 P^2 * [P]_k = [P]_{k+1},
%F A113350 P^2 * [Q]_k = [Q]_{k+1},
%F A113350 Q^2 * [P^2]_k = [P^2]_{k+1},
%F A113350 Q^2 * [Q^2]_k = [Q^2]_{k+1},
%F A113350 for all k>=0.
%e A113350 Triangle Q begins:
%e A113350 1;
%e A113350 2,1;
%e A113350 5,4,1;
%e A113350 19,22,6,1;
%e A113350 113,166,51,8,1;
%e A113350 966,1671,561,92,10,1;
%e A113350 10958,21510,7726,1324,145,12,1;
%e A113350 156700,341463,129406,23010,2575,210,14,1;
%e A113350 2727794,6496923,2572892,471724,53935,4434,287,16,1;
%e A113350 56306696,144856710,59525136,11198006,1305070,108593,7021,376,18,1;
%e A113350 Matrix square Q^2 begins:
%e A113350 1;
%e A113350 4,1;
%e A113350 18,8,1;
%e A113350 112,68,12,1;
%e A113350 965,712,150,16,1;
%e A113350 10957,9270,2184,264,20,1; ...
%e A113350 where Q^2 transforms column k of Q^2 into column k+1:
%e A113350 at k=0, [Q^2]*[1,4,18,112,965,...] = [1,8,68,712,9270,...];
%e A113350 at k=1, [Q^2]*[1,8,68,712,9270,...] =
%e A113350 [1,12,150,2184,37523,...].
%o A113350 (PARI) Q(n,k)=local(A,B);A=matrix(1,1);A[1,1]=1;for(m=2,n+1,B=matrix(m,m); for(i=1,m, for(j=1,i,if(i<3 || j==i || j>m-1,B[i,j]=1,if(j==1, B[i,1]=1,B[i,j]=(A^(2*j-1))[i-j+1,1]));));A=B);(A^(2*k+2))[n-k+1,1]
%Y A113350 Cf. A113351 (column 1), A113352 (column 2), A113353 (column 3), A113354 (column 4); A113355 (Q^2), A113365 (Q^3), A113340 (P), A113345 (P^2), A113360 (P^3).
%K A113350 nonn,tabl
%O A113350 0,2
%A A113350 _Paul D. Hanna_, Nov 08 2005